Equity warrant pricing under subdiffusive fractional Brownian motion of the short rate
aa r X i v : . [ q -f i n . P R ] J u l EQUITY WARRANT PRICING UNDER SUBDIFFUSIVEFRACTIONAL BROWNIAN MOTION OF THE SHORT RATE
FOAD SHOKROLLAHI
Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700,FIN-65101 Vaasa, FINLAND
MARCIN MAGDZIARZ
Hugo Steinhaus Center, Department of Applied Mathematics, Wroc law Universityof Science and Technology, Wyspia´nskiego 27, Wroc law, 50-370, Poland
Abstract.
In this paper we propose an extension of the Merton model. Weapply the subdiffusive mechanism to analyze equity warrant in a fractional Brow-nian motion environment, when the short rate follows the subdiffusive fractionalBlack-Scholes model. We obtain the pricing formula for zero-coupon bond in theintroduced model and derive the partial differential equation with appropriateboundary conditions for the valuation of equity warrant. Finally, the pricing for-mula for equity warrant is provided under subdiffusive fractional Black-Scholesmodel of the short rate. Introduction
Analysis of financial data displays that various processes viewed in finance showcertain periods in which they are constant [14]. Analogous property is observedin physical system with subduffusion. The constant periods of financial processescorrespond to the trapping event in which the subdiffusive particle is motionless[18, 19, 6]. The mathematical interpretation of subdiffusion is in terms of Frac-tional Fokker Planck equation (
F F P E ). This equation was introduced from thecontinuous time random walk (
CT RW ) strategy with fat tail waiting times [18, 19],later used as a substantial tool to evaluate complex system with slow dynamics. Inthis paper we use the fractional Black-Scholes (
F BS ) model and the subdiffusivemechanism to better describe the dynamics observed in financial markets. We usesimilar strategy as in [17, 25], where the objective time t was replaced by the in-verse α -stable subordinator T α ( t ) in the F BS model. T α ( t ) corresponds to thefat-tailed waiting times in the underlying CT RW and adds the constant periodsto the dynamics of financial assets. Then, the dynamic of asset price V ( t ) is givenby the following subdiffusive F BS
E-mail addresses : [email protected], [email protected] . Date : July 27, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Merton short rate model; Subdiffusive processes; Fractional Brownianmotion; Equity warrant. dV ( T α ( t )) = µ v V ( T α ( t )) d ( T α ( t )) + σ v V ( T α ( t )) dB H ( T α ( t )) , (1.1)where µ v , σ v are constant, B H is the fractional Brownian motion ( F BM ) [1]with Hurst parameter H ∈ [ , T α ( t ) is the inverse α -stable subordinator with α ∈ (0 ,
1) defined as T α ( t ) = inf { τ > U α ( τ ) > t } , (1.2)Here { U α ( t ) } t ≥ is a α -stable L´evy process with nonnegative increments andLaplace transform: E (cid:0) e − uU α ( t ) (cid:1) = e − tu α [15, 8, 28, 10]. When α ↑ T α ( t )degenerates to t . t V ( t ) Figure 1.
Typical trajectory of the asset price from formula (1.1).The parameters are: µ v = σ v = V (0) = 1 , H = 0 . , α = 0 . dr ( T α ( t )) = µ r d ( T α ( t )) + σ r dB H ( T α ( t )) . (1.3)Here µ r , σ r are some constants, B H is a F BM with Hurst parameter H ∈ [ , T α ( t ) is assumed to be independent of B H . Moreover, B H and B are twodependent F BM s with correlation coefficient ρ . Additionally T α ( t ) is assumedto be independent of B H and B H . RACTIONAL SHORT RATE 3 t r( t ) Figure 2.
Typical trajectory of the short rate from formula (1.3)corresponding to the trajectory from Fig. 1. The parameters are: µ r = σ r = r (0) = 1 , H = 0 . , α = 0 . , ρ = 0 . , F , Q ) on which the fractional Brownianmotion is defined, we present some “ideal conditions” in the market for the firmvalue and for the equity warrant:(i) There are no transaction costs or taxes and all securities are perfectlydivisible;(ii) Dividends are not paid during the lifetime of the outstanding warrants,and the sequential exercise of the warrants is not optimal for warrantholders;(iii) The warrant-issuing firm is an equity firm with no outstanding debt; SHOKROLLAHI AND MAGDZIARZ (iv) There are no signaling effects associated with issuing warrants. The mar-ket neither reacts positively nor negatively to the information that thefirm will issue (or has issued, respectively) warrants;(v) Suppose that the firm has N shares of common stocks and M shares ofequity warrants outstanding. Each warrant entitles the owner to receive k shares of stocks at time T upon payment of X .Assumptions (i)–(ii) are the standard assumptions in the Black–Scholes envi-ronment. Assumption (iii) implies that stocks and equity warrants are the onlysources of financing that are issued by the firm. Assumption (iv) is a necessarycondition for our analysis. This assumption implies that both the stock price andthe firm’s value remain unaffected by warrants issuance. Actually, this conditioncould be achieved, see [11]. Assumption (v) sets some notations for the pricingmodel. Throughout this research, we assume that α ∈ ( ,
1) and α + αH > Pricing model for a zero-coupon bond
In this section, we assume that the short rate r ( t ) satisfy Equation (1.3). Then,we obtain the pricing formula for zero-coupon bond P ( r, t, T ). Here, P ( r, T, T ) =1, that is, the zero coupon bond will pay 1 dollar at expiry date T . Theorem 2.1.
The price of a zero-coupon bond with maturity t ∈ [0 , T ] in thefractional Black-Scholes model is given by P ( r, t, T ) = e − rf ( τ )+ f ( τ ) , (2.1) where τ = T − t and f ( τ ) = Hσ r (Γ( α )) H Z τ ( T − v ) ( α − H +2 H − v dv − Hµ r (Γ( α )) H Z τ ( T − v ) ( α − H +2 H − vdv, (2.2) f ( τ ) = τ. (2.3) Proof.
By the Taylor series expansion, we can get P ( r + ∆ r, t + ∆ t ) = P ( r, t, T ) + ∂P∂r ∆ r + ∂P∂t ∆ t + 12 ∂ P∂r (∆ r ) + + 12 ∂ P∂r∂t ∆ r (∆ t ) + 12 ∂ P∂t (∆ t ) + O (∆ t ) . (2.4)From Eq. (1.3) and [8] and [21], we have∆ r = µ r (∆ T α ( t )) + σ r B H ( T α ( t ))= µ r (cid:18) t α − Γ( α ) (cid:19) (∆ t ) + σ r ∆ B H ( T α ( t )) + O (∆ t ) . (2.5) (∆ r ) = σ r (cid:18) t α − Γ( α ) (cid:19) H (∆ t ) H + O (∆ t ) . (2.6) ∆ r (∆ t ) = O (∆ t ) . (2.7) RACTIONAL SHORT RATE 5
Hence dP ( r, t, T ) = " µ r t α − Γ( α ) ∂P∂r + σ r Ht H − (cid:18) t α − Γ( α ) (cid:19) H ∂ P∂r + ∂P∂t dt + σ r ∂P∂t dB H ( T α ( t )) . (2.8)Putting µ = 1 P " µ r t α − Γ( α ) ∂P∂r + σ r Ht H − (cid:18) t α − Γ( α ) (cid:19) H ∂ P∂r + ∂P∂t ,σ = 1 P (cid:18) ∂P∂r (cid:19) , (2.9)and letting the local expectations hypothesis holds for the term structure of interestrates (i.e. µ = r ), we have ∂P∂t + µ r t α − Γ( α ) ∂P∂r + Ht H − σ r (cid:18) t α − Γ( α ) (cid:19) H ∂ P∂r − rP = 0 . (2.10)Then, zero-coupon bond P ( r, t, T ) with boundary condition P ( r, t, T ) = 1 satisfiesthe following partial differential equation ∂P∂t + µ r t α − Γ( α ) ∂P∂r + Ht H − σ r (cid:18) t α − Γ( α ) (cid:19) H ∂ P∂r − rP = 0 . (2.11)To solve Equation (2.11) for P ( r, t, T ), let τ = T − t, P ( r, t, T ) = exp { f ( τ ) − rf ( τ ) } , then we can get ∂P∂t = P (cid:18) − ∂f ( τ ) ∂τ + r ∂f ( τ ) ∂τ (cid:19) , (2.12) ∂P∂r = − P f ( τ ) , (2.13) ∂ P∂r = P f ( τ ) . (2.14)Replacing Equations (2.13) and (2.14) into Equation (2.12) and simplifying Equa-tion (2.11) we get P " Ht H − σ r f ( τ ) (cid:18) t α − Γ( α ) (cid:19) H − µ r f ( τ ) t α − Γ( α ) − ∂f ( τ ) ∂τ + r (cid:18) ∂f ( τ ) ∂τ − (cid:19) = 0 . (2.15) SHOKROLLAHI AND MAGDZIARZ
From Equation (2.15), we obtain ∂f ( τ ) ∂τ = σ r Ht H − (cid:18) t α − Γ( α ) (cid:19) H f ( τ ) − µ r t α − Γ( α ) f ( τ ) ,∂f ( τ ) ∂τ = 1 . (2.16)Then, f ( τ ) = Hσ r (Γ( α )) H Z τ ( T − v ) αH − v dv − µ r Γ( α ) Z τ ( T − v ) α − vdv, (2.17) f ( τ ) = τ. (2.18)This ends the proof. (cid:3) T P (r ,t, T ) H=0.6H=0.7H=0.8
Figure 3.
Bond price as a function of maturity time T for differentvalues of H , see formula (2.1). The parameters are: µ r = σ r = r (0) = 1 , α = 0 . , t = 0. Corollary 2.1.
When α ↑ , Equations (1.1) and (1.3) reduce to the F BS . Thenwe obtain f ( τ ) = Hσ r Z τ ( T − v ) H − v dv − µ r Z τ vdv, (2.19) specially, if t = 0 f ( τ ) = σ r T H +2 (2 H + 1)(2 H + 2) − µ r T , (2.20) then RACTIONAL SHORT RATE 7 P ( r, t, T ) = exp (cid:26) − rT + σ r T H +2 (2 H + 1)(2 H + 2) − µ r T (cid:27) . (2.21) Corollary 2.2. If H = , from Equation (2.17), we obtain f ( τ ) = 12 σ r Γ( α ) Z τ ( T − v ) α − v dv − µ r Γ( α ) Z τ ( T − v ) α − vdv, (2.22) then the result is consistent with the result in [9] .Further, if α ↑ and H = , Equations (1.1) and (1.3) reduce to the geometricBrownian motion, then we have f ( τ ) = 16 σ r τ − µ r τ , (2.23) then P ( r, t, T ) = e − rτ + σ r τ − µ r τ , (2.24) which is consistent with the result in [16, 4] . Fractional Black-Scholes equation
This section provides corresponding
F BS equation for equity warrants whenthe stock price and short rate satisfy Eqs. (1.1) and (1.3), respectively. Recallthat B H and B H are two dependent F BM with Hurst parameter H ∈ [ ,
1) andcorrelation coefficient ρ . Theorem 3.1.
Let assumptions (i)–(v) hold and W t be the valuation of the equitywarrant. Then the valuation of equity warrant at time t ∈ [0 , T ] satisfies thefollowing P DE and the following boundary condition ∂W∂t + e σ v ( t ) V ∂ W∂V + e σ r ( t ) ∂ W∂r + 2 ρ e σ r ( t ) e σ v ( t ) ∂ W∂V ∂r + µ r t α − Γ( α ) ∂W∂r + rV ∂W∂V − rW = 0 , (3.1) with boundary condition W T = 1 N + M k ( kV T − N X ) + (3.2) where e σ v ( t ) = Ht H − σ v (cid:18) t α − Γ( α ) (cid:19) H , (3.3) e σ r ( t ) = Ht H − σ r (cid:18) t α − Γ( α ) (cid:19) H . (3.4) σ v , σ r , µ r , µ v , are constant, H ∈ [ , and α ∈ ( , . SHOKROLLAHI AND MAGDZIARZ
Proof.
It is worth pointing out that W t is a function of the current value of theunderlying asset V t , the stochastic interest rate r t and time t : W t = W ( V t , r t , t ).Then, at the expiry date t = T , we have W T = 1 N + M k ( kV T − N X ) + , (3.5)which is the boundary condition of Eq 3.1. Consider a portfolio with one shareof warrants, D t shares of stock and D t shares of zero-coupon bond P ( r, t, T ).Then the value of the portfolio Π at current time t isΠ t = W t − D t V t − D t P t . (3.6)Then, from [9] we have d Π t = dW t − D t dV t − D t dP t = " ∂W∂t dt + Ht H − σ v V t (cid:18) t α − Γ( α ) (cid:19) H ∂ W∂V + Ht H − σ r (cid:18) t α − Γ( α ) (cid:19) H ∂ W∂r + 2 Ht H − ρσ r σ v V (cid:18) t α − Γ( α ) (cid:19) H ∂ W∂V ∂r dt + " ∂W∂t − D t dV t + " ∂W∂r − D t ∂P∂r dr + D t " ∂P∂t + Ht H − σ r (cid:18) t α − Γ( α ) (cid:19) H ∂ P∂r dt. (3.7)Set D t = ∂W∂V , D t = ∂W∂r∂P∂r , to eliminate the stochastic noise. Then d Π t == " ∂W∂t + Ht H − (cid:18) t α − Γ( α ) (cid:19) H (cid:18) σ v V ∂ V∂V + σ r ∂ W∂r + 2 ρσ r σ v V ∂ W∂V ∂r (cid:19) dt − ∂W∂r∂P∂r " rP − µ r t α − Γ( α ) ∂P∂r dt. (3.8)The return of the amount Π t invested in bank account is equal to r ( t )Π t dt attime dt , E ( d Π t ) = r ( t )Π t dt = r ( t ) ( W t − D t V t − D t P t ) , hence from Equation(3.8) we have ∂W∂t + Ht H − (cid:18) t α − Γ( α ) (cid:19) H (cid:18) σ v V ∂ C∂S + σ r ∂ W∂r + 2 ρσ r σ v V ∂ W∂V ∂r (cid:19) + µ r t α − Γ( α ) ∂W∂r + rv ∂W∂V − rW = 0 . (3.9)Let e σ v ( t ) = Ht H − σ v (cid:18) t α − Γ( α ) (cid:19) H , (3.10) e σ r ( t ) = Ht H − σ r (cid:18) t α − Γ( α ) (cid:19) H . (3.11) RACTIONAL SHORT RATE 9
Then ∂W∂t + e σ v ( t ) V t ∂ W∂V t + e σ r ( t ) ∂ W∂r + 2 ρ e σ r ( t ) e σ v ( t ) ∂ W∂V ∂r + µ r t α − Γ( α ) ∂W∂r + rV ∂W∂V − rW = 0 , (3.12)and the proof is completed. (cid:3) Theorem 3.2.
Let assumptions (i)–(v) hold and W t be the valuation of the equitywarrant. Then the valuation of an equity warrant at time t ∈ [0 , T ] is given by W t = 1 N + M k (cid:16) kV t φ ( d ) − N Xe − r ( T − t ) P ( r, t, T ) φ ( d ) (cid:17) , (3.13)(3.14) where d = ln kV t NX − ln P ( r, t, T ) + H (Γ( α )) H R Tt b σ ( v ) v αH − dv q H (Γ( α )) H R Tt b σ ( v ) v αH − dv , (3.15) d = d − s H (Γ( α )) H Z Tt b σ ( v ) v αH − dv, (3.16) b σ ( t ) = σ v + 2 ρσ r σ v ( T − t ) + σ r ( T − t ) . (3.17) P ( r, t, T ) is given by Eq (2.1) and φ ( . ) is the cumulative standard normal distri-bution function.Proof. Consider the partial differential equation (3.1) of the equity warrants withboundary condition W T = N + Mk ( kV T − N X ) + ∂W∂t + e σ v ( t ) V ∂ W∂V + e σ r ( t ) ∂ W∂r + 2 ρ e σ r ( t ) e σ v ( t ) ∂ W∂V ∂r + µ r t α − Γ( α ) ∂W∂r + rV ∂W∂V − rW = 0 , (3.18)Denote z = VP ( r, t, T ) , Θ( z, t ) = W ( V, r, t ) P ( r, t, T ) . (3.19)Then we get ∂W∂t = Θ ∂P∂t + P ∂ Θ ∂t − z ∂ Θ ∂z ∂P∂t ,∂W∂r = Θ ∂P∂r − z ∂ Θ ∂z ∂P∂r ,∂W∂V = ∂ Θ ∂z , (3.20) ∂ W∂r = Θ ∂ P∂r − z ∂ Θ ∂z ∂ P∂r + z P ∂ Θ ∂z (cid:18) ∂P∂r (cid:19) ,∂ W∂r∂V = − zP ∂ Θ ∂z ∂P∂r ,∂ W∂V = 1 P ∂ Θ ∂z . Inserting Equation (3.20) into Equation (3.18) ∂ Θ ∂t + ∂ Θ ∂z "e σ v ( t ) V P + 2 ρz e σ r ( t ) e σ v ( t ) 1 P ∂P∂r + e σ r ( t ) z (cid:18) P ∂P∂r (cid:19) − zP (cid:20) ∂P∂t + e σ r ( t ) ∂ P∂r + µ r t α − Γ( α ) ∂P∂r − r Vz (cid:21) + Θ P (cid:20) ∂P∂t + e σ r ( t ) ∂ P∂r + µ r t α − Γ( α ) ∂P∂r − rP (cid:21) = 0 . (3.21)From Equation (2.11), we can obtain ∂ Θ ∂t + σ ( t ) z ∂ Θ ∂z = 0 , (3.22)with boundary condition Θ( z, T ) = ( z − K ) + ,where σ ( t ) = e σ v ( t ) + 2 ρ e σ r ( t ) e σ v ( t )( T − t ) + e σ r ( t ) ( T − t ) . (3.23)The solution of partial differential Equation (3.22) with boundary conditionΘ( z, T ) = N + Mk ( kz − N X ) + , is given byΘ( z, t ) = kzφ ( b d ) − N Xφ ( b d ) , (3.24)here b d = ln kzNX + R Tt σ ( v ) ds q R Tt b σ ( v ) dv , (3.25) b d = b d − s Z Tt σ ( v ) dv. (3.26)Thus, from Equations (3.19) and (3.24)–(3.26) we obtain W ( V, r, t ) = 1 N + M k (cid:16) kV t φ ( d ) − N Xe − r ( T − t ) P ( r, t, T ) φ ( d ) (cid:17) (3.27) RACTIONAL SHORT RATE 11 where d = ln kV t NX − ln P ( r, t, T ) + H (Γ( α )) H R Tt b σ ( v ) v αH − dv q H (Γ( α )) H R Tt b σ ( v ) v αH − dv , (3.28) d = d − s H (Γ( α )) H Z Tt b σ ( v ) v αH − dv. (3.29) (cid:3) T W t H=0.6H=0.7H=0.8
Figure 4.
Value of equity warrant as a function of maturity time T for different values of H , see formula (3.14). The parameters are: µ v = σ v = µ r = σ r = r (0) = V (0) = N = M = k = X = 1 , α =0 . , ρ = 0 . , t = 0 . Acknowledgements
The research of MM was partially supported by NCN SONATA BIS-9 grantnumber 2019/34/E/ST1/00360.
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